The two exercises I'd like help with is on the picture, thanks (:
Hello, Fnus!
Here's the first one . . .
The point (-2,3) is on the graph.has a stationary point at
(a) Find the values of and
(b) Find the position and nature of all stationary points.
. . Hence: . .[1]
Since (-2,3) is a stationary point,
. . and we have: .
Substitute into [1]: .
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
The function is: .
Then: .
When . . . Critical point:
When . . . Critical point:
Second Derivative Test: .
. . . concave up . . . minumum at
. . . concave down . . . maximum at
Here's 6
Let
Now, let's piece our clues together.
Clue 1: touches the line at the point
we can take two things away from this.
(1) we interpret the phrase "touching the line" to mean, the line is tangent to the curve. thus it means that
(2) the second thing we can take from this, is that passes through the point , that is which means of course,
two unknowns down, two to go.
what else do we know about ?
Clue 2: has a stationary point at
again, there are two things we can take from this.
(1) passes through , that is,
that means that: (i plugged in the values we got for c and d)
simplifying, we get:
......................(1)
(2) the second thing we take from this is that (that is, the slope of when is zero)
but that means that: (i plugged in the value for c)
simplifying we get:
......................(2)
thus, to find the remaining unknowns, we must solve the system
......................(1)
......................(2)
the solutions are and
thus,
EDIT: i made an error. when i concluded that c = 2, i claimed the wrong thing, i should have said that (since 9 is the slope of the line). make the necessary corrections, i can't bother